# Centers of motion

Bodies with space-time orbit by gravitation around their barycenter, the center of mass. The word barycenter is from the Greek βαρύς, heavy + κέντρον, center. The barycenter is one of the foci of the elliptical orbit of each body.

For the two-body case let m and M be the two masses, and let r and R be vectors to m and M respectively. Then the center of mass or barycenter is

(mr + MR) / (m + M).

Define the reduced mass μ = mM/(m + M). Then the orbit is as if the orbiting body has reduced mass μ and there is a stationary central body with mass equal to the total mass (m + M). That is, the two bodies mutually orbit the center of mass.

Let’s reconsider the orbit in relation to the elaphrances, the mass inverses, orbiting by levitation. For the two-movement case let  and L be the two elaphrances, and let r and R be vectors from an origin to and L respectively. Then the center of elaphrance is

(ℓr + LR)/( + L) = (Mr + mR) / (m + M).

Define the reduced elaphrance Λ = ℓL/( + L). Then the orbit is as if the orbiting movement has reduced elaphrance λ and there is a stationary central movement with elaphrance equal to the total elaphrance ( + L). That is, the two movements mutually orbit the center of elaphrance.

The result for elaphrance is the same except that the roles of the movements are reversed. One could call the center of elaphrance the elaphrocenter after ελαφρός, light (unheavy) + κέντρον, center.